We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Annual review of condensed matter physics

The concept of a continuous module is a generalization of that of an injective module, and conditions (), (C) and () are given for this concept in [4]. In this paper, we study modules with properties that are dual to continuity. These will be called discrete and we discuss discrete abelian groups. Throughout R is a ring with identity, M is a module over R, G is an abelian group of finite rank, E is the ring of endomorphisms of G and S is the center of E. Dual to the notion of essential submodules, we define small submodules of a module M over R.(omitted)

On discrete groups

Sur le probléme des courbes gauches en Topologie

Complex networks have been studied intensively for a decade, but research still focuses on the limited case of a single, non-interacting network. Modern systems are coupled together and therefore should be modelled as interdependent networks. A fundamental property of interdependent networks is that failure of nodes in one network may lead to failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. A dramatic real-world example of a cascade of failures (‘concurrent malfunction’) is the electrical blackout that affected much of Italy on 28 September 2003: the shutdown of power stations directly led to the failure of nodes in the Internet communication network, which in turn caused further breakdown of power stations. Here we develop a framework for understanding the robustness of interacting networks subject to such cascading failures. We present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Surprisingly, a broader degree distribution increases the vulnerability of interdependent networks to random failure, which is opposite to how a single network behaves. Our findings highlight the need to consider interdependent network properties in designing robust networks.

http://absimage.aps.org/image/MAR10/MWS_MAR10-2009-001087.pdf

Catastrophic cascade of failures in interdependent networks

Gromov hyperbolic graphs

On the hyperbolicity constant in graphs

If X is a geodesic metric space and x"1,x"2,x"[email protected]?X, a geodesic triangle T={x"1,x"2,x"3} is the union of the three geodesics [x"1x"2], [x"2x"3] and [x"3x"1] in X. The space X is @d-hyperbolic (in the Gromov sense) if any geodesic side of T is contained in a @d-neighborhood of the union of the two other geodesic sides, for every geodesic triangle T in X. We denote by @d(X) the sharpest hyperbolicity constant of X, i.e. @d(X):=inf{@d>=0:X is @d-hyperbolic}. In this paper we prove that in order to compute the hyperbolicity constant in a graph with edges of the same length, it suffices to consider geodesic triangles such that the three points determining those triangles are vertices of the graph or midpoints of edges of the graph. By using this result we prove that the hyperbolicity constant of a graph with edges of length k is a multiple of k/4.

Computing the hyperbolicity constant

Spectral properties of geometric-arithmetic index

Let G = (V,E) be a graph of order n and let B(S) be the set of vertices in
 V\S that have a neighbor in the vertex set S. The differential of a vertex
 set S is defined as ∂(S) = |B(S)| - |S| and the maximum value of ∂(S) for
 any subset S of V is the differential of G. A Roman dominating function of G
 is a function f : V → {0,1,2} such that every vertex u with f(u) = 0 is
 adjacent to a vertex v with f(v) = 2. The weight of a Roman dominating
 function is the value f(V) = ∑uV f(u). The minimum weight of a Roman
 dominating function of a graph G is the Roman domination number of G,
 written γR(G). We prove that γR(G) = n - ∂(G) and present several
 combinatorial, algorithmic and complexity-theoretic consequences thereof.

/pdf/the-differential-and-the-roman-domination-number-of-a-graph-1xpkptl0yn.pdf

José M. Sigarreta

Papers

Gromov hyperbolic graphs

On the hyperbolicity constant in graphs

Computing the hyperbolicity constant

Spectral properties of geometric-arithmetic index

The differential and the roman domination number of a graph